NETWORK SYNTHESIS 655 



of the known relation (100), connecting N and D. Let E and be re- 

 spectively the sums of even and odd terms in N. Then A'' is E + and 

 (100) re(|uires D to be E - 0. The ratio 0/E may be related to X) ^kz' 

 of (98) as follows : 



I = -tanh I E C,z' (103) 



Now let two convergent series, respectively even and odd, be such 

 that: 



- = -tanhi^ZC,^' (104) 



Let coefficients Kk be defined by: 



J^ + = E ^i^^' " (105) 



Then E and are respectively the sums of the even and odd terms. 

 The complete series may now be related to the (odd) series ^ Ckz'^ as 

 follows : 



S hCz = -log E K'uz' (106) 



This fixes the Kk oi E -{- in terms of the Ck . 



To make Ck = Ck through m odd orders, (102) is now replaced by 



""" , , 



E = B (10^) 



mo 



The symbol = designates equality of power series through m odd orders. 

 {All even terms are now zero on both sides.) The right hand side may be 

 expressed as a continued fraction of the following form, comparable to 



(86): 



(108) 



Truncation after only the m^^ term gives the 0/E of (107). 



The coefficients of and E may be calculated by an appropriate 

 modification of (61). (Calculate like Kk of (101), after dividing all Ck 

 by 2). After and E have been evaluated, by truncating the continued 

 fraction (108), their sum gives polynomial A'' of (99). 



